// Removes a separator from the graph, and recursively computes a tree decomposition. The separator should be safe.
// If check3isolated is true, then this also checks the property that given a separator of size 3, it should not split off any isolated vertices (or else it is not safe).
// If any connected component has more than maximumSize vertices, do not proceed (balanced separator check)
public Graph RemoveSeparator(List <Vertex> separator, bool check3isolated, int maximumSize)
{
// Compute the graph after removing separator
List <Vertex> tempVertices = vertices.Keys.ToList();
foreach (Vertex v in separator)
{
tempVertices.Remove(v);
}
// Get the connected components
Graph componentsGraph = new Graph(tempVertices);
List <List <Vertex> > components = componentsGraph.getComponents();
// If applicable, check that it is safe
if (check3isolated && components.Any((c) => c.Count == 1))
{
return(null);
}
// Check balance
if (components.Any((c) => c.Count > maximumSize))
{
return(null);
}
// Prepare the new tree decomposition: the root bag contains the separator
Graph result = new Graph();
Program.BagsList.Add(new List <Vertex>(separator));
Vertex root = result.AddVertex(Program.BagsList.Count - 1);
// Decompose each component
foreach (List <Vertex> component in components)
{
// Create a new graph with the component in it
Graph componentGraph = new Graph(component);
// Add the separator
foreach (Vertex v in separator)
{
componentGraph.AddVertex(v);
}
foreach (Vertex u in separator)
{
// Make the separator a clique
foreach (Vertex v in separator)
{
if (u.Label <= v.Label) // Not strictly necessary, but might save some time
{
continue;
}
componentGraph.AddEdge(u, v);
}
// And add edges from the separator to the rest
foreach (Vertex v in u.Adj)
{
componentGraph.AddEdge(u, v);
}
}
// Compute the TD of the component
Graph TDComponent = componentGraph.Decompose();
result.AddGraph(TDComponent);
// Find an appropriate bag in the decomposition to connect our root bag to
foreach (Vertex v in TDComponent.vertices.Keys)
{
if (separator.All((u) => { return(Program.BagsList[v.Label].Contains(u)); }))
{
result.AddEdge(root, v);
break;
}
}
}
return(result);
}
// Recursively computes a tree decomposition
// Tries several reduction rules (connected components, size 1,2,3 separators, almost clique separators)
public Graph Decompose()
{
Graph result = new Graph();
// Most simple case: the graph is disconnected
getComponents();
if (components.Count > 1)
{
Vertex prevVertex = null;
foreach (List <Vertex> component in components)
{
Graph componentGraph = new Graph(component);
Graph TDComponent = componentGraph.Decompose();
// Combine TDs into path (any way that makes the components connected is OK
result.AddGraph(TDComponent);
Vertex TDVertex = TDComponent.vertices.First().Value;
if (prevVertex != null)
{
result.AddEdge(TDVertex, prevVertex);
}
prevVertex = TDVertex;
}
return(result);
}
// Biconnected graph: An articulation point is always safe (single vertex separator)
Vertex ap = ArticulationPoint();
if (ap != null)
{
return(RemoveSeparator(new List <Vertex>(new Vertex[] { ap })));
}
// 3-connected graph: try to find an articulation pair by removing a vertex, then testing if some other vertex is an articulation point
// These are always safe
// Note that thia can be done more efficiently, but this is easier and fast enough for reasonable graphs
foreach (Vertex v1 in vertices.Keys)
{
List <Vertex> tempVertices = vertices.Keys.ToList();
tempVertices.Remove(v1);
Vertex v2 = new Graph(tempVertices).ArticulationPoint();
if (v2 != null) // Found an articulation pair (v1, v2)
{
return(RemoveSeparator(new List <Vertex>(new Vertex[] { v1, v2 })));
}
}
// 4-connected graph: same idea, remove pair of vertices, then find an articulation point
// Again quite inefficient (cubic) but we've already done a lot of reduction so the graph is hopefully small at this point
foreach (Vertex v1 in vertices.Keys)
{
foreach (Vertex v2 in vertices.Keys)
{
if (v1.Label >= v2.Label)
{
continue;
}
List <Vertex> tempVertices = vertices.Keys.ToList();
tempVertices.Remove(v1);
tempVertices.Remove(v2);
Vertex v3 = new Graph(tempVertices).ArticulationPoint();
if (v3 != null) // Found a separator (v1, v2, v3)
{
// RemoveSeparator checks that this is a safe separator (not all 3-vertex separators are safe) and returns null if not
Graph temp = RemoveSeparator(new List <Vertex>(new Vertex[] { v1, v2, v3 }), true);
if (temp != null)
{
return(temp);
}
}
}
}
// Our final reduction rule: a minimal separator that is a clique or almost clique (a clique plus a vertex) is safe
List <List <Vertex> > cliques = new List <List <Vertex> >();
foreach (Vertex v in vertices.Keys) // Initialization: all vertices cliques of size 1
{
cliques.Add(new List <Vertex>(new Vertex[] { v }));
}
// Enumerate all cliques; we don't need to consider cliques that are as large as the entire graph
for (int i = 2; i < vertices.Count - 1; i++)
{
List <List <Vertex> > newCliques = new List <List <Vertex> >();
foreach (List <Vertex> clique in cliques) // Enumerate over cliques of size i-1
{
foreach (Vertex v in vertices.Keys)
{
if (v.Label <= clique[clique.Count - 1].Label)
{
continue; // Optimization: only consider cliques where the vertices are in increasing order
}
if (clique.All((u) => { return(edges.Contains(new Edge(u, v))); })) // Can we add the vertex (is it connected to everything else)?
{
// Found a new clique
List <Vertex> newClique = new List <Vertex>(clique);
newClique.Add(v);
newCliques.Add(newClique);
// Copy the graph, remove a clique
List <Vertex> tempVertices = vertices.Keys.ToList();
foreach (Vertex u in newClique)
{
tempVertices.Remove(u);
}
// If the graph minus the clique has an AP, the clique plus the AP is an (almost) clique minimal separator
Vertex v1 = new Graph(tempVertices).ArticulationPoint();
if (v1 != null) // Found an (almost) clique separator
{
newClique.Add(v1);
return(RemoveSeparator(newClique));
}
}
}
}
cliques = newCliques;
// Limit the number of cases considered, as we want to limit the time and don't necessarily need to consider all of them
cliques.Shuffle();
cliques = cliques.Take(10000).ToList();
}
// No more reduction rules apply: we now try to find a tree decomposition using a 2^n DP algorithm
return(ComputeTD());
}